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Commutative rings with toroidal zero-divisor graphs. (English) Zbl 1226.05095

Summary: Let \(R\) be a commutative ring and let \(\Gamma (R)\) denote its zero-divisor graph. We investigate the genus number of a compact Riemann surface in which \(\Gamma(R)\) can be embedded and explicitly determine all finite commutative rings \(R\) (up to isomorphism) such that \(\Gamma (R)\) is either planar or toroidal.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13M05 Structure of finite commutative rings
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